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Geometry of Large Random Trees: SPDE Approximation

  • Yuri BakhtinEmail author
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2068)

Abstract

In this chapter we present a point of view at large random trees. We study the geometry of large random rooted plane trees under Gibbs distributions with nearest neighbour interaction. In the first section of this chapter, we study the limiting behaviour of the trees as their size grows to infinity. We give results showing that the branching type statistics is deterministic in the limit, and the deviations from this law of large numbers follow a large deviation principle. Under the same limit, the distribution on finite trees converges to a distribution on infinite ones. These trees can be interpreted as realizations of a critical branching process conditioned on non-extinction. In the second section, we consider a natural embedding of the infinite tree into the two-dimensional Euclidean plane and obtain a scaling limit for this embedding. The geometry of the limiting object is of particular interest. It can be viewed as a stochastic foliation, a flow of monotone maps, or as a solution to a certain Stochastic PDE with respect to a Brownian sheet. We describe these points of view and discuss a natural connection with superprocesses.

Keywords

Random Tree Large Deviation Principle Stochastic Partial Differential Equation Gibbs Distribution Brownian Sheet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 8.
    Aldous, D.: The continuum random tree. II. An overview. In: Barlow, M.T., Bingham, N.H. (eds.) Stochastic Analysis. London Mathematical Society Lecture Note Series, vol. 167. Cambridge University Press, Cambridge (1991)Google Scholar
  2. 34.
    Bakhtin, Y.: Thermodynamic limit for large random trees. Random Struct. Algorithm 37, 312–331 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 35.
    Bakhtin, Y.: SPDE approximation for random trees. Markov Process. Relat. Fields 17, 1–36 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 37.
    Bakhtin, Y., Heitsch, C.: Large deviations for random trees. J. Stat. Phys. 132, 551–560 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 38.
    Bakhtin, Y., Heitsch, C.: Large deviations for random trees and the branching of RNA secondary structures. Bull. Math. Biol. 71, 84–106 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 69.
    Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley Series in Probability and Statistics: Probability and Statistics. Wiley, New York (1999)Google Scholar
  7. 164.
    Dynkin, E.B.: Markov processes. Vols. I, II. Die Grundlehren der Mathematischen Wissenschaften, Band 121/122. Academic, New York (1965)Google Scholar
  8. 166.
    Ellis, R.S.: Entropy, large deviations, and statistical mechanics. Classics in Mathematics. Springer, Berlin (2006). Reprint of the 1985 originalGoogle Scholar
  9. 168.
    Ethier, S.N., Kurtz, T.G.: Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1986)Google Scholar
  10. 169.
    Evans, S.N.: Two representations of a conditioned superprocess. Proc. Roy. Soc. Edinb. Sect. A 123, 959–971 (1993)zbMATHCrossRefGoogle Scholar
  11. 170.
    Evans, S.N., Perkins, E.: Measure-valued Markov branching processes conditioned on nonextinction. Israel J. Math. 71, 329–337 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 266.
    Itô, K., McKean, H.P. Jr.: Diffusion Processes and Their Sample Paths. Springer, Berlin (1974). 2nd printing, corrected, Die Grundlehren der mathematischen Wissenschaften, Band 125Google Scholar
  13. 285.
    Karatzas, I., Shreve, S.E.: Methods of mathematical finance. Applications of Mathematics (New York), vol. 39. Springer, New York (1998)Google Scholar
  14. 294.
    Kesten, H.: Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Stat. 22, 425–487 (1986)MathSciNetzbMATHGoogle Scholar
  15. 316.
    Kunita, H.: Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, vol. 24. Cambridge University Press, Cambridge (1997). Reprint of the 1990 originalGoogle Scholar
  16. 480.
    Stanley, R.P.: Enumerative combinatorics. Vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999)Google Scholar
  17. 505.
    Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’été de probabilités de Saint-Flour, XIV—1984. Lecture Notes in Mathematics, vol. 1180, pp. 265–439. Springer, Berlin (1986)Google Scholar
  18. 531.
    Zuker, M., Mathews, D.H., Turner, D.H.: Algorithms and thermodynamics for RNA secondary structure prediction: A practical guide. In: Barciszewski, J., Clark, B.F.C. (eds.) RNA Biochemistry and Biotechnology. NATO ASI Series. Kluwer, Dordrecht (1999)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Georgia Institute of TechnologyAtlantaUSA

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